A general inner-outer iteration for computing extreme eigenpairs of symmetric/positive-definite matrix pencils is proposed. The principle of the method is to produce a sequence of p-dimensional bases {Xk} that converge to a minimizer of a generalized Rayleigh quotient. The role of the inner iteration is to produce an “update” vector by (approximately) minimizing a quadratic model of the Rayleig...

We investigate the influence of the inner profile of the lens objects on gravitational lensing statistics taking into account of the effect of the magnification bias and both the evolution and the scatter of halo profiles. We take the dark matter halos as the lens objects and consider the following three models for the density profile of the dark matter halo; SIS (singular isothermal sphere), t...

This work is concerned with the development and study of a minimum residual norm subspace method based on the generalized conjugate residual method with inner orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of nonsymmetric or non-Hermitian linear systems. First we recall the main features of flexible generalized minimum residual with...

In this paper we extend the definition of generalized Sobolev space and subsequent theoretical results established recently for positive definite kernels and differential operators in the article [21]. In the present paper the semi-inner product of the generalized Sobolev space is set up by a vector distributional operator P consisting of finitely or countably many distributional operators Pn, ...

In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction [25] for a generalized eigenvalue problem. The method applies ...

The preconditioned inverse iteration [Ney01a] is an efficient method to compute the smallest eigenpair of a symmetric positive definite matrix M . Here we use this method to find the smallest eigenvalues of a hierarchical matrix [Hac99]. The storage complexity of the datasparse H-matrices is almost linear. We use H-arithmetic to precondition with an approximate inverse of M or an approximate Ch...

2 A denote the transpose, Moore-Penrose generalized inverse, Frobenius norm and Euclid norm, respectively. For any , m n A B  R ,   , 0 T A B trace B A   denotes the inner product of A and B . Therefore, m n R  is a complete inner product space endowed with 2 , A A A  . For any non-zero matrices 1 2 , , , m n k A A A  R  , if  , T j i i A A trace A     0 j A i j   , then it is...

The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationsh...

Recent results have shown that any closed operator A commuting with the backwards shift S∗ restricted to K u := H 2 ⊖ uH, where u is an inner function, can be realized as a Nevanlinna function of S∗ u := S ∗| K2 u , A = φ(S∗ u), where φ belongs to a certain class of Nevanlinna functions which depend on u. In this paper this result is generalized to show that given any contraction T of class C0(...

The effect of an equilibrium sheared ow on the nonlinear evolution of a neoclassical tearing mode is investigated by estimating the inuence it has on the inner and outer layer dynamics of the mode. Two complementary approaches are adopted. A generalized Rutherford model calculation is carried out to estimate the ow contributions to the polarization current term in the inner layer. For the ou...